SEPTEMBER 1985 ANDREW C. VASTANO AND ROBERT O. REID 393Sea Surface Topography Estimation with Infrared Satellite Imagery ANDREW C. VASTANO AND ROBERT O. REIDDepartment of Oceanography, Texas A&M University, College Station, TX 77843(Manuscript received 20 July 1984, in final form 30 November 1984)ABSTRACT Sea surface flow derived from displacements of surface patterns in sequential NOAA-6 AVHRR (11 micronband) satellite images yield coherent nonuniform distributions of velocity vectors. An analytic representationof flow over the region of the distribution is obtained by performing a least-squares regression analysis fo/coefficients ofa streamfunction expansion that is expressed in terms of trigonometric basis functions. Sea surfacetopography is estimated with the streamfunction by employing a geostrophic approximation. An application ismade to a portion of the Oyashio Frontal Zone in the northwestern Pacific that includes the First and SecondOyashio Intrusions and an anticyclonic eddy. A horizontal map of a local rotational perturbation property iscalculated for this region as a further example of the use of the streamfunction analysis.1. Introduction Analyses of discretely sampled velocity fields forstreamfunction representations of flow were carried outby Pritchard (1948) to obtain monthly average surfacecurrents on basinwide scales. Streamline characterizations for flow regimes in the Caribbean Sea as well asportions of the eastern North Pacific and the Gulf ofMexico were derived from charts compiled with shipreports of surface currents. In contrast, the present interest is in single mesoscale features and mesoscalefields and requires intensive observational efforts togather synoptic data sets applicable to flow studies.Field experiments must often employ multiplatformsurveys and instrument arrays to achieve samplingdensities that permit resolution of significant structuraland temporal changes. Near-synoptic current data asin Hubertz et al. (1972) or hydrographic data as inMcWilliams (1976) are essential, sampled at horizontalspatial and temporal resolutions in the orders of tensof kilometers and days. Satellite orbital characteristicsand sensor resolutions provide such sampling for seasurface expressions of mesoscale phenomena with infrared, microwave and visible instrumentation. In addition to present detection and tracking applications,satellites have an enormous potential for providingquantitative dynamic information about mesoscalefields and supplementing surface-acquired observations. Estimates of motion for mesoscale features and theirgross movement have been made with infrared imageryby several investigators (e.g., Vastano and Bernstein,1984). A more recent study (Vastaho and Borders,1984) produced a field of sea surface flow vectors formesoscale elements on the Oyashio Front northeast ofJapan. An interactive algorithm requires sequential infrared (AVHRR) (11 micron band) images and com-.putes surface velocity vector components from useridentified displacements of sea surface temperaturepatterns and elapsed time. This procedure assumes thatthe total time rate 'of change of a scalar quantity depends solely on horizontal advective processes and isdemonstrated to have a measured repeatability of 1-2cm s-I for successive images approximately 24 hoursapart. Comparison of the velocity results with surfaceacquired flow measurements made during a previousfield experiment indicates a general agreement in speedand direction over a semipermanent anticyclonic eddynear the thermal ridge of the Oyashio Front. Satellitederived vector fields of this nature suggest streamfunction estimates that are made for nonuniform vectordistributions as contrasted to the uniformly distributedvector fields used by Pritchard (1948). In turn, thestreamfunctions can permit assessments of sea surfacetopographies that are independent of microwave altimetric observation. The results reported in this paperprovide a method for extracting mesoscale sea surfacetopography from sequential infrared images and are astep toward extrapolating altimetric topography datato mesoscale areal coverage.2. Surface topography The use of sequential infrared images to extract surface motion estimates (Vastano and Borders, 1984)produces a nonuniform distribution of velocity components u (+ eastward) and v (+ northward) that represent advective movements apparent in the changesof sea surface temperature patterns. The advectivemovements of small perturbations in the temperaturepattern are estimated by an investigator using an interactive image processing algorithm. The velocityc 1985 American Meteorological Society394 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOI.,OGY VOLUME2vectors (Fig. 1) are a sample of the motion that can beconsidered approximately planar since vertical motionsare nominally much smaller than horizontal ones. Atemperature range of approximately 1 I-C is represented in the grey scale distribution of the image (Vastano and Bernstein, 1984). The perturbations in thefrontal regions associated with the warm core eddy aredefined by four degree differences. The premise is thatthese features are created by small-scale wave or turbulent processes, represent perturbations of otherwisesmooth surface isotherms, and reveal the flow regimeby their displacements. Specifically, the features are ofsmall scale relative to the baroclinic radius of deformation. Although Rossby-like or internal gravity wavescan be present, it follows that their phase speeds aremuch less than maximum theoretical values and aretherefore regarded as slower than typical mesoscalesurface currents. Diffusion, heat exchange with the atmosphere and vertical motions are assumed to producechanges of temperature much less than those inducedby advective processes for sequential image time intervals. A mathematical expression of this concept canbe approach~ed through the material derivative of ascalar quantity C,DC OC--=--+V. VC=R. (1)Dt OtC has mean (Co) and pertufi>ation (Ci) componentsand the flow regime is composed of mean (V0) andperturbation (V0 components. Thus C=Co+Gand V = Vo + V~, (2)where Vo is presumed nearly parallel to contours ofCo. The mean components of C and - are consideredto be slowly evolving relative to rapid spatial.and temporal fluctuations that characterize the perturbations.The right-hand side of (1) represents processes thatcause negligible changes in the scalar quantity C overthe time interval between images. Expanding (1) withthe components of C and V yields FIG. 1., Surface flow vectors from 20 and 21 May 1981 NOAA-6, AVHRR (11 micron band)images and sea surface topography ovedayed on the 20'May image. Tick marks are placed at twodegree latitude and longitude intervals.SEPTEMBER 1985 ANDREW C, VASTANO AND ROBERT O. REID 395whereOCt-- -- V0 - VCi -~- Si q- 82, (3)Ot OCo S~ = R - Vo - VC0 - VI - VC1 - -- (4) Dt s2 =-v~ - vc0. (5) Equation (4) includes the self-advective or turbulent diffusion term of the flow regime; S~ is disregarded as a result of the short time interval. Equation (5) rep resents an episodic source term that gives rise to the' perturbations in the scalar field. Between events, the relation OC~ 0--~- + Vo - VC~ = 0 (6)remains to relate the scalar and flow fields. The Lagrangian equivalent of (6) states that the velocities -0of small water parcels are equivalent to the time rateof change of their displacements. The vectors shownin Fig. 1 are computed from such movements and areassumed to be governed by Eq. (6). When this flow isconsidered horizontally nondivergent, the ocean surface velocity field is derivable from a single scalar function, the streamfunction ,It. Under an approximationof uniform Cofiolis parameter, the streamfuncfion isproportional to the surface topography and the flowrendition is a geostrophic one. One way of deriving. ~ from the velocity field is tointerpolate the velocity data. to a rectangular grid(McWilliams, 1976). This facilitates estimates of vorticity from which xI, can be obtained by solution ofPoisson's equation as in Hubertz et al. (1972). A simpler method is adopted here that avoids prior interpolation of the observed velocity field and minimizesthe problem of specifying boundary conditions iuherent in the solution of Poisson's equation. The methodassumes that the field of - can be represented over thedata set by a series of selected basis functions in x (east),y (north). The coefficients of the series are evaluatedby a least-squares fit to the velocity data by the appropriate derivatives of ,I~. Representing the streamfunction in terms of trigonometric basis functions, mrx m~ry 9 = ~ ~ A~,, sin-- sin , - m ' Lx L~velocity component estimates have the form(7) 09 m~r . mrx m~ryt~=-~y =-~ m~-~VA~'"s'n-c-s-' Lx L~ (8) 0~ mr A cosnXX . m~y = Lx sin- ~ (9)where the x, y coordinates are scaled by Lx and Ly, thedimensions of the rectangular region enclosing the datasamples. The selection of sine terms for the streamfunction implies that - vanishes on an enlarged rectangular boundary surrounding the velocity vector distribution. A least-squares error function tr2 = [Z (/J- Uj)2 q- (17- l)j)2lljmax (10) Jover the set of observations uy, vi, permits a variationon the coefficients and results in a linear set of equations [Z(/, k)][A(k)] = [F(/)] (11)for the coefficients, where k and l = n + (m - 1) nmaxand Z(l, k) = Y. [(atak)i + (btb0~] (12) J Hl) = Y. (atu~ + btv)and, in general,(13) mr . nrx m~y = ---sm--cos-- (14)a Ly Lx Ly rn ~r y mr cosn.~x sin (15)b = - L~ L~ LyTwo constraints are added to the system that requirethe individual sums of (t7 - ui) and (ff - vi) to vanish.The solution for - is accomplished using standardGauss-Jordan elimination procedures for inversion ofthe Z(I, k) matrix. The relation of the streamfunctionto ea surface topography is made by invoking a geostrophic approximation for the flow such thatV- 0xlt- g 0n (16) Os fOs'wherefis the Coriolis parameter, g is the accelerationof gravity,, and the coordinate s is taken normal tostreamlines.and contours of n, the sea surface topography. In this case, integration yieldsconstant, (17)wherefis taken as a constant appropriate to the centroid of the data distribution. The sea surface topography analysis constrains thesatellite-observed flow components to approximate thesurface relief in geostrophic terms. Clearly these components contain more information in addition to thebaroclinic contribution to the flow field. The time interval between images is chosen to be short relative tothe scalar changes induced by a number of mesoscaleprocesses. However, the analysis is not comprehensiveeven under this selection. Wind forcing of the surface396 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOL.OGY VOLUME2layer has not yet been accounted for in the analysis.Such an inclusion is feasible over an image area onlywith wind observations at spatial and temporal resolutions comparable to the flow vector field and wouldrequire a sensor similar to the Seasat (SASS). Otherdynamic mechanisms, such as centripetal acceleration,can be pertinent depending on the types of mesoscalefeatures present in the images. Comparisons of satellitederived velocities with surface-acquired measurementsfor eddies and plumes are in progress. Results of thesestudies will provide information relevant to the necessity for more complex analyses. At present, a statistical measure of experimental repeatability for the topography can be quantified byconsidering an ensemble of possible data realizationsfrom the same image pair that have equivalent velocitycomponent variances. It follows from (7) and (17) thatdepartures (7') of ~ from its expected value for a givenx, yare = - ~' A' . mrx m~ry (18) 7' f Z ~-~ n,m Sln--z"ffx sin' L-~-' g n mwhere A~.m represent departures ofAn, m from expectedvalues for the ensemble. The expected value of themean square deviation (MSD) for the topography isgoverned by = Z Z Z ~.~n,m p,qjj \~1 n m p q [. n~rxj . p~rxjsln~' m~ryjsinq~ryj~ (19)x ~,sin--~ sln--~-ff~ Ly ~ 'IfA~,m and A~,,e are stochastically independent realizations from an A ensemble whose variance is independent of n and m, then , , = ~(A')2; k = n, 1 = m (20)An'mAp'q [0 ; k = n or l = m,andZj (73')2 = (~)2~At)2 n~ m~ j~. (sin2 -~xtl~rxj sin2 ~m~ryj~]' (21)A similar analysis for deviations of the computed velocity components (u, v) gives ~ [(u3)2 + (v3)2] = (A')2 ~ ~ (ak2 + bk2)~ = (A')2 trZ. (22)Substitution for (A')~ in (21) yields a working relationfor the expected MSD of ~ over the data set.1 Z (,/~)2 = t_~Jmax jx ~ ~ ~[s~n'-~x sin2m~rYJ~Ly ]where the term in braces is identified with the righthand side of (10), the variance (a2) of the fitted velocityto the data sample. The vector distribution shown in Fig. I was obtainedfrom sequential NOAA-6 AVHRR (11 micron band)images taken over a 23.6 hour interval on 20-21 May1981 (Vastano and Borders, 1984). The images encompass a geographic region slightly larger than thearea defined by 38-44-N, 144.-150-E. The central feature is a semipermanent anticyclonic eddy between theFirst and Second Oyashio Intrusions located southeastof the Japanese island of Hokkaido. The sample of theflow field is comprised of 115; vector estimates of advective motion within the eddy, along the thermal ridgeof the Oyashio Front, and wi~Ihin the Tohoku mixing'zone to the south. A sequence of numerical experiments were carded out in which the order of the trigonometric basis functions was increased. The variance~2 steadily decreased with ascending orders andasymptotically approached a limiting value. This limiting variance is a measure of the departure of the velocity data sample from geoslrophy. Figure 2 presentsa scatter plot of observed versus calculated velocitycomponents derived from a slreamfunction expansionof seventh order in north and. east basis functions andhas an associated a2 of 10.1 cm2 s-2 and a goodness offit (r2) of 0.81. The computed topographic relief forthis approximation is shown in Fig. 3. The dimensionsof the rectangular outline are slightly smaller than thefitted region chosen to include the vector distribution(L~, = 446 km, Lv = 362 km). The expected MSD forthe topography values evaluated from (23) is 0.30 cm;this is a measure of experime:ntal repeatability and nota measure of error. The central anticyclonic eddy hasan excursion slightly greater than +20 cm, indicatinga topographic dome relative to a -15 cm contour tothe north across the thermal ridge. Altimeter oceansurface topography profiles a,ver warm core eddies arenot available for the northwestern Pacific. However,using Seasat altimetry, Marslh et al. (1982) found thathigh mesoscale variability is present in the geographicregion containing the warm core eddy and gives a rangeof rms surface height variability of 16-22 cm. The satellite topographic relief does correspond well to thatfound by Cheney and Marsh ( 1981) for warm core ringsgenerated by the Gulf Stream in the North AmericanSEPTEMBER 1985 ANDREW C. VASTANO AND ROBERT O. REID 397 c(LEBS,UC~.) , A(VQSS,VC~LJ - L FIG. 2. Scatter plot of observed vs calculated velocity components (cm s-t). Observed valuesare those obtained from. the sequential images. Calculated values are computed from thestreamfunction.Slope Water region. A series of Seasat altimeter passesthat intercepted warm core rings showed domes rangingfrom 75 (pass 277-A) to 45 (pass 486-D) cm relativeto background. A single XBT measurement taken on15 May 1981 in the anticyclonic eddy of Fig. I showedsubarctic waters (<4-C) below 100 m depth and warmwater (> 10-C) at the surface (Vastano and Bernstein,1984). This shallow temperature perturbation relativeto the environment suggests that a smaller topographicrange should be expected in comparison to the NorthAtlantic rings sampled by Seasat and indicates that thetopography shown in Fig. 3 gives a reasonable estimateof the sea surface variation. The sea surface topography or streamfunction contours over the warm core eddy in Fig. I show a nearlycircular and symmetric field that represents a horizontally nondivergent approximation of the local flowpattern. Interest in the perturbations from a circularmean state and eddy azimuthal structure (Olson andSpence, 1978; Mied et al. 1983; McCalpin, 1984) hasbeen generated by their evident relation to eddy instabilities and the availability of high spatial resolutionfield experiments and numerical models. While theseinvestigations used field data in the form of temperaturemaps and numerically computed layer heights, Olson(1980) discussed the relevance of horizontal potentialvorticity gradients to the baroclinic instability of mesoscale eddies. Vertical hydrographic sections wereutilized since shipboard sampling limitations do notpermit quasi-synoptic horizontal maps of vorticity. Afurther example of the quantitative information derivable from satellite flow estimates can be given by extracting the horizontal surface distribution of a localrotational property from the vector field in Fig. 1. Thisproperty is estimated by considering the quantity Q= x72xP - -r2~ which, in the context of geostrophicflow, is proportional to the perturbation potential vorticity defined by Gill (1982). The term Q is expressedin the vertical normal mode form where 3~ is the inverseof the radius of deformation for the first baroclinic eigenmode. The property Q + ~y is approximately conserved following a fluid parcel where ~ is df/dy. Hydrographic data were not taken over the eddy in theMay 1981 period and the radius of deformation (40km) was established on the basis of historical verticalsections of warm core eddies in the region (Tomosada,1975). Figure 4 presents contours of Q(107) S-1 overlayed on the vector distribution. The Q field near the398JOURNALOF ATMOSPHERICAND OCEANIC TECHNOLOGYVOLUME 2FIG. 3. Sea surface tOl~ography (cm) computed with the least-squares method overlaye~l on thevelocity vectors shown in Fig. 1. Dots are placed at the tail of each vector.center shows closed negative (anticyclonic) contoursover the eddy with high values skewed to the westernside and a broad, low value region extending to thenortheast. There are no synoptic surface acquired vorticity maps for comparison to Fig. 4. Olson (1980) hasgiven a zonal potential vorticity section that bisects acyclonic eddy and shows slight departures from radialsymmetry. Numerical experiments such as those reported by Smith and O'Brien (1983) have showntongues of eddylike vorticity advected around eddiesthat have shapes similar to the northeastern extensionof the eddy in Fig. 4. The Q field does have shape,magnitudes and features over the eddy that are reasonable in terms of previous results. The values nearthe edge of the map and Ihe closure of the outlyingfeatures must be regarded as artifacts of the analysisprocedure that imposes a zero streamfunction value atthe boundaries.3. Discussion The topography in Fig. 3 is an estimate of the seasurface distribution to within the constant of integration in (17). Information about the mesoscale motionis present, since flow estimates are obtained from derivatives of this scalar field. The integration constantcan be evaluated by computing dynamic heights fromhydrographic stations or by microwave satellite observations. The primary disadw~ntage in the implementation of this method is the necessity for cloud-free3iews of the sea Surface. Active microwave sensing ofthe sea surface such as that obtained by Seasat providestopographic information that is relatively free of thisproblem; yet' its application to mesoscale studies ishampered by narrow (3 kin) sensor footprints andtemporal resolution. Movement of oceanic features andthose orbital repetition rates tlhat produce "gappy" datawill result in aliasing for mesoscale interpretations ofthe altimetric data. In addition, the precision of altimeters is presently approxima'~ely 10 cm, a figure considerably larger than the attainable experimental repeatability of the least-squares method. The altimetererror in derived topography will be reduced with bettergeoid definition and a combination of the least-squaresmethod and altimetric obsep~ation is clearly indicated.SEPTEMBER 1985 ANDREW C. VASTANO AND ROBERT O. REID 399I,,_,,&-ZF~G. 4. Perturbation vorticity property Q(IO?) s-~ computed from streamfunction and overlayed on the velocity vectors shown in Fig. 1. The application of the least-squares method providesmesoscale flow fields that have areal coverage sufficientto define the sea surface expressions of individual mesoscale features and surroundings with temporal resolution at possible half-day intervals. This informationcan supplement data gathered in field experiments bysurface platforms and can significantly enhance theirinterpretation. In fact, velocity estimate sources otherthan infrared imagery such as hydrography, drifters,GEK observations, visible imagery and radar altimetrycan be included to provide data sets for the least-squaresmethod. The topographic fields or derivative velocityfields present the opportunity to initialize and verifymesoscale computer models and can be employedalone for local forecasts of environmental flow conditions. As an example of an extension of the method,the velocity vector fields can be combined with coincident, accurate sea surface temperature maps of similar resolution such as those derived from satellite datawith corrections for atmospheric cloud and water vaporeffects (Bernstein, 1982; Vastano and Bernstein, 1984).The combination will provide estimates of the totaltime rate of change of surface temperature and advective flow for studies of the local time rate of change oftemperature in mesoscale fields and will be applicable'to heat content studies of the oceanic surface layer. Acknowledgments. The image processing was carriedout at the Satellite-Oceanography Facility, Scripps Institution of Oceanography, La Jolla, California. Theauthors thank Robert Whritner for assistance in archiving the images of the Oyashio Front and KorenAbdullah for help in the flow field analysis. AndrewVastano was supported by Grant OCE 80-26037 of theNational Science Foundation and the Office of NavalResearch under Contract N00014-75-0537.REFERENCESBernstein, R. L., 1982: Sea surface temperature estimation using the NOAA-6 satellite AVHRR radiometer. J. Geophys. Res., 87, 9455-9465.Cheney, R. E., and J. G. Marsh, 1981: Seasat altimeter observations of dynamic topography in the Gulf Stream region. J. Geophys. Res., 86, 473-483.400 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME2Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, 662 pp. Hubertz, J. M., A. W. Garcia and R. O. Reid, 1972: Objective analysis of oceanic surface currents. 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